Research Activities

In the Laboratory of Evolution Equations investigations in the following main directions are carried on:

• General theory of initial boundary value problems for linear equations and systems
  of elliptic, parabolic, and quasi-elliptic types,
  including the questions of existence of classical solutions,
  solutions estimating in uniform norms,
  boundary value problems in domains with a nonsmooth boundary.

• Theory of nonlinear evolutionary equations of parabolic and hyperbolic types,
  questions of solvability of basic initial boundary value problems
  “on the whole” with respect to time,
  asymptotic behavior of solutions, their stabilization and stability,
  attraction domains of stable stationary or periodic solutions,
  theorems on integral manifolds.

• Lyapunov methods in the problem of stability of solutions
  to linearized parabolic and hyperbolic systems,
  localization of the spectrum of corresponding differential operators,
  and solvability of hereby arising operator equations.

• Application of the spectral theory of differential operators to solving inverse problems
  for evolutionary equations of mathematical physics.

• General questions of calculus of variations and its applications to solution
  of problems of nonlinear theory of elasticity.

• Spectral properties of operators arising in the theory
  of small oscillations of rotating ideal fluid
  and dependence of qualitative characteristic features of solutions
  on the character of the spectrum of the corresponding problem.